nf0934: holography is a very modern part in optics it's very complicated subject as well as interesting it's possibly one of the most interesting things m-, most of the interesting topics that we have discussed in the lecture er and er it is going to be in a very simplified manner what i'm going to show you so don't be surprised if y-, if i'm just if i just throw formulas at you er that are are the final result don't expect any complicated mathematics the mathematics in holography are very very complex so the intention that i have today with holography is to just explain to you what it is er make you understand the basic principles of holography and make you feel comfortable with the idea so that's the intention there isn't much math but there are some very interesting though complex ideas regarding waves in optics and er i would ask you to pay a little bit of attention and try to think on the way okay i am sure that you all have heard before the word holography you know that it is technique that produces three-dimensional images and er the question is what is it that's different in holography compared to two-dimensional photographs what is that we miss when we've got a normal photograph or we've got our television screen what we see is just a two-dimensional flat image we don't see depth basically what we see is just irradiance so light intensity goes on the screen it gets reflected and that information goes into our eye and we see a two-dimensional image but we miss something the what we miss is information and that information that we miss is the phase of the reflected wavefront as we all know a wave can be described by two things can be described by the amplitude which effectively defines the irradiance and the phase so in two dimensions what we see is simply irradiance but we don't see phase with techniques of holography what we see is both irradiance and phase and this is how we end up seeing a three-dimensional image now the question is that if we want to see both the irradiance and the phase we need to have a mechanism from which we can code that information and when we talk about phase then sometimes it's easy to think interference and basic idea of holography is to encode both information phase and amplitude in form of interference fringes later on we'll see how we can achieve that right and what i'm going to show you now after i just made a short introduction is give you a schematic representation of what happens in holography holography is defined er is separated sorry into two parts one part is the recording of the information the other part is the reconstruction of the information so keep that balance so you can keep good control on what's being said in the lecture right now here i'm going to show you two figures oops i don't want to ruin that [laughter] so we've got two figures here i do not can you can you see it clearly is it okay okay no complaints so i can understand that you can see everything clearly now let's see what we've got here before to those who didn't hear it i said that the basis of holography is storing information about both amplitude and phase and let's see what happens we've got here a beam that is broad and it's coming from a laser that beam is split then in two one part of the beam gets reflected on a mirror and shines on a photographic plate so that's one part of the recording the other part of the beam carries on shines on the object that we're studying and then it gets reflected back on the photographic plate so we've got two beams coming from a laser that travel different paths and what do you think is going to happen on that photographic plate when these two beams come together there are two beams that are coherent because they come from a laser and they travel different paths and i'm sure you all know by now after the exam that you did all your homework that what we have is interference these two beams are going to interfere and what happens when they interfere we're expecting on this photographic plate to have a complex set of dark and bright fringes that's the result of interference and take my word for it at the moment that these fringes are a coded form of the information that you need to have a three-dimensional reconstruction of that object these fringes tell you everything you need to know if you want to have a three- dimensional reconstruction of that object we'll see later the explanation for that right so then when we develop that photographic film we say that we can reconstruct the actual object how that is illustrated on the second figure here and what we have is a laser beam again shining on that photographic plate and these fringes that are there on the photographic plates make an a complex wave to arise on the other side of the plate and believe me at this stage that if you look at an angle through that photographic plate you will see a three- dimensional reconstruction of the original object so that is the basic idea the very simplistic idea behind holography now what i'm going to do is i'm going to try to explain to you the mechanisms of holography using two different ways one is a pictorial Fourier way and the second one is going to be a direct mathematical simple way they're both important to understand so i'd like to ask you to pay attention to both er ways of dealing with the problem my first explanation is going to be based on Fourier analysis or Fourier optics i'm not quite sure how many of you feel familiar with Fourier analysis can you just put your hands up the ones that feel comfortable with it have you heard of it before at all you have er er anybody else is it totally new to you Fourier analysis hands up to those that don't know anything about Fourier analysis ooh okay okay now because i wasn't certain which of you know things and which don't i've added an extra some extra information regarding basic things about Fourier analysis most of them when they most the people when they hear about Fourier analysis they panic they think it's too complicated but if you just try to concentrate on the idea i'm sure that you will all be better than experts so what is Fourier analysis then Fourier analysis is a way to describe an image to describe a signal to describe a function it's a language nothing more or less than that for that particular case when we talk for images we say that Fourier analysis provides the tools to analyse an image how i'll show you first the horrid thing the mathematics so that's what it is c-, can you see it i think it's quite oops i don't see that surviving [laughter] okay so what's the first formula the first formula shows you how a function F- of-X-and-Y with that particular function and that particular case is an image is an intensity can be analysed in Fourier spatial frequencies of a particular amplitude and phase you must have seen that that integration before i mean it is very popular here is in two dimensions because we're talking about two-dimensional images so a scene can be expressed as a series of some Fourier coefficients the coefficients are characterized with a capital letter the spatial frequencies are U and V the integration is over the spatial frequencies i'll tell you later what a spatial frequency is so don't don't be demotivated and here is again another expression where i show how these Fourier coefficients can be expressed can be calculated when you know the scene F-of-X so that's basically the mathematical idea the relation between the scene and something that we call Fourier coefficients the important thing about Fourier coefficients is spatial frequency have you heard that term before spatial frequency okay so now let's forget the math and see in practice what we mean so what i've got here is a very simple image on the right which is a step what i've got here is let's say high intensity what i've got here is no intensity so it's high intensity dark high intensity dark and so on it's a simple function the reason why i picked a simple function is because it's easier to represent but what holds for that particular function holds for the complex image so it's the same thing now to make it more simple what i've done is i've taken one line out of that image one line across that image and that line is shown here as a one- dimensional image and what i want you to see now is how we can reconstruct that sort of complex intensity line using Fourier analysis and that result is shown on the last figure here and what i have is the square of intensity and on top i've drawn a simple sine function then on top of that basic simple sine function i've drawn another sine function of different frequency and on the last figure here if you've got more than two sine functions and these sine functions when you add them up you're going to have their initial intensity reconstructed so what is Fourier analysis it's a way to break a complex scene into simpler things that you can play with that you can work with what simple sine functions of different amplitudes of different wavelength when you add all these sine functions what's the idea it's to reconstruct your one-dimensional image so that's Fourier analysis and it's shown here in one dimension for a simple image and on the figure above what i'm showing is one sine function shown in one dimension and here it is how that function shows on two dimensions that represents intensity yeah so that's the idea behind Fourier analysis now what i'm wanting you to concentrate if you want to understand the rest is what is spatial frequency that's very important and that is shown on the figure below so what i've done on the figure below is i have taken one of these sine functions and i've plotted it out and i'm saying that the period of that sine function is T the period is T T is not time just a period of that time function what is its spatial frequency is one over-T that's what it is that's a spatial frequency and spatial frequency can be defined linearly as one- over-T or angularly as i've shown on that part of the figure where the difference with angular spatial frequency is that it's important from which position you're looking at something so you've got this period of that sine function you're sitting there looking at that sine function and that period is making an angle theta with you with the observer the angular spatial frequency is one- over- theta so what i'm wanting you to remember out of that transparency is that a scene can be analysed in spatial frequencies these spatial frequencies are a way to reconstruct a more complex scene and why is it important to analyse it it is because as the light falls on an object and then gets reflected there is a theory in optics that say that what travels actually after the light has been reflected is the spatial frequencies of that particular scene and each spatial frequency is travels on a wave and depending on th-, how big that spatial frequency is no the angle that the spatial frequency makes with the central axis is higher for higher spatial frequencies so what i'm saying is that maybe i need to use blackboard for that one what i'm saying is that when you have light reflected off that scene you've got that's the central axis you've got a spatial frequency travelling let's say straight through which is the zero order but but don't worry about that at the moment you've got another spatial frequency travelling like this you've got another one travelling like this and so on and they propagate these spatial frequencies and because of the lens they just come back together on the image plane they recombine and there you see the final image okay so that's something that we need to remember how the light propagates in terms of spatial frequencies now why is it important is shown and understood from the following transparency so going back to the dea-, the idea of holography we said that we've got two parts of a laser beam recombining on a photographic plate and we said that there we've got fringes on that photographic plate and what i'm trying to go er what i'm going to do now is to show you the result of taking one spatial frequency and see the result of that spatial frequency with the original laser beam and if i put up the figure it's going to be easier for you to follow right so what do i have here this is going to be an attempt to explain to you how the image is recorded with holography yeah so i am taking just one spatial frequency of the scene yeah that particular spatial frequency is shown here in in it is assumed that it makes an angle theta with the reference wave which is the part of the beam that comes straight from the laser and is shown on the photographic plate right the straight lines represent crests and the dotted lines represent troughs and that is our photographic plate and i need to hurry and that is our photographic plate and the reference beam is assumed in this figure to have a maximum on that photographic plate now what i'd like you to do is to tell me if you can see at which points on the photographic plate i'm going to have maxima occurring from interference between the two beams sm0935: A B and C nf0934: that's excellent this is where you have your maxima A B and C now what is going to happen to the light intensity in between these points sm0936: nf0934: it will depend on the phase difference between the two beams the classic line from interference so the next step is is to actually mathematically try and calculate how much that phase difference is going to be as a function of position on top of that photographic plate that's our next aim that's what we need to do and now i've got well another figure which is basically part of the first one but only showing the interesting bits for the calculation so again i've got my photographic plate there i've got an axis X on that photographic plate and i've got the angle theta that the spatial frequency makes with the reference wave got the wavelength of my radiation and what i'm wanting to do is to calculate the phase phi phi how do you call it phi yes phi along that photographic plate now you're not supposed to see that bit yet [laugh] so how much do you think that phase difference is going to change if i travel from B to A where the two maxima occur sm0937: two-pi nf0934: exactly so when i go from the two maxima the phase has changed by two- pi therefore taking that into consideration one can write that the phase phi at some point X satisfies that relationship that analogy if you want but phi over two-pi equals X over the length A-B don't worry making notes about that because you've got everything in your handout so you don't you needn't worry about it unless of course if it helps you learn things better to which is which is fine by me and now what we want to do is to isolate that phi and we say that the phase is two- pi multiplied by X divided by A-B that calculation though is not finished because what because what we actually want to relate is that phase difference with the angle theta that the spatial frequency makes with the reference wave and how do we do that usual way we use that triangle yeah and from that triangle we're going to substitute the length A-B that is not very helpful to us at the moment and it's quite straightforward to see that the sine of that angle theta is the wavelength lambda over A-B therefore that length A-B is going to be the length sorry the wavelength divided by the sine of the wanted angle therefore our final result is that the phase along the photographic plate as a function of position is going to be two-pi over the wavelength multiplied by the distance multiplied by the sine of the angle that the spatial frequency makes with the reference wave and now what is it important to see the calculation is quite straightforward i'm sure you all understood it the important is that the phase X that defines if we're going to have intensity maxima or minima is not only a function of position X it is a function of the angle that the spatial frequency makes with the reference wave and what did we say before that a complex scene can be broken into a number of spatial frequencies that travel with different angles where do i want to end i want to say that for each spatial frequency that angle theta is going to be different therefore what we're expecting to see on the photographic plate when you've got a complex object is a very complicated form of of of dark and bright fringes that's what we're expecting to see but before we go to that point the next bit is to see what is going to be the light field on that photographic plate and the light field on that photographic plate is going to be given by this formula that comes to you like out of the blue now but it really comes from interference the point is not to go through all the steps it's to make you understand what's behind holography yeah and that's the resultant wave of two beams that have got the same amplitude now what's the resultant intensity when the field is like that the resultant intensity as we all know is going to be proportional to the square of the wave time average blah blah and believe me that is going to be given by that equation so what do we see that for one spatial frequency the intensity on the photographic plate is expected to have a constant term and another term who has got a cosine dependence cosine dependence on the phase difference and enough about math how's it going to look like i'm going to show you here how the fringes are going to look like on that photographic plate for this simple case and we said that it's going to look like a cosine and indeed it looks like that which is a cosine function okay so that's now the intensity on the photographic plate and what do we do when we have that information we develop the film and then we say okay now we've got our fringes we want to reconstruct what's the idea behind reconstruction that is very simply shown on that one where you've got now another laser shining on that photographic plate and because you've got intensity variations on that photographic plate you've got a complex wavefront arising behind that plate and you've got a number of spatial frequencies travelling in different directions and what i'm saying is that if one looks from that position he's going to see a three-dimensional object the explanation about reconstructing is going to be from my point of view more successful with the other approach with a direct mathematical approach but that's the end of the first way of dealing with holography the Fourier way how are the fringes going to look for a more complicated object they're going to look more complex and they are bound to look more complicated the more complicated the object you are studying is so i'm just showing you here two pictures to see how do they look like one finds it hard to believe that out of that you've got so much information but that's how it is in holography two more points that i need to make is that as we said before when you have a more complex object you're expecting the phase difference phi to give you a more complicated configuration of fringes and if the amplitude of the waves is not the same as we said before then that is going to reflect on how bright these fringes are we'll see more details of that one on the second part of how to work with holography which is something that you may feel more comfortable which is direct mathematical way so we're going back to the usual old known ideas about waves and to remind you again of the idea of holography i'm just going to very briefly show you what we said about recording we said we've got a laser beam it's been broadened part of it shines on that photographic plate and another part of the beam shines on the object and then ends up on the photographic plate so what do we've got we've got two electric fields recombining on a photographic plate usual interference problem right so what do we know about the usual interference problem we just write down the two parts of the wave in a simple mathematical form so what's the first equation represent the first equation represents the part of the laser beam that has not been reflected off from the object and that part of the beam we call it the background beam or reference beam and this is why you've got the substr-, subscript- B for background beam that's the electric field that is being produced on the photographic plate and is of constant amplitude and of phase that is a function of position on the photographic plate that's the description of the first part of the beam now we've got the beam that has been scattered by the object that we give the name E- O O stands for object and that is again the amplitude which is now a position of i-, it's a function sorry of position on the photographic fil-, on the photographic film and a f-, er a phase phi with the subscript- O for object which is again a function of position on the photographic field and what we want to do is to calculate the total electric field that arises from these two fields overlapping adding with each other and what we do is we say that the field is going to be the addition of these two and the intensity that it is what we are worried about is going to be as we all know the field squared and then time averaged and the result of that calculation is going to be as you all know the amplitude of the background beam squared plus the amplitude of the object beam squared plus that interesting bit here which is a cosine of the difference in phase these two beams have now never mind about amplitudes and about constants that's that's the key point here so we say then we see that the intensity formed on that photographic film is really a function of the difference in phase between these two beams so what can we say that the intensity is a coded form of the phase difference that's the whole point that i want you to remember on that one right we understood now the importance of the phase that the phase has been recorded on the photographic film but what about the other bit of information which is the intensity and we're saying that the intensity in holography defines the contrast as i said to you before in the scene and that contrast V is the maximum-intensity-minus-the-minimum-intensity over the maximum-intensity-plus- the-minimum that's a simple definition of what contrast is and that contrast relates if you want to amplitude by this simple definition and we see that the amplitude of the object E-O-O indeed defines the contrast of these fringes so the contrast and the intensity really of that photographic plate give you all the information you need both information for intensity and phase to reconstruct your original scene in three dimensions right so that's the bit about recording for the direct mathematical approach what about reconstruction again i'm showing you very briefly the figure for those that didn't pay that much attention before you've got your reference beam on reconstruction that shines on that photographic film that has all the information encoded and part of the beam is going through and the wavefront that arises behind that photographic field ph-, photographic film sorry is giving you the information you need to see that three-dimensional image now one may say hold on a minute you've got two images there you've got one here and one there and indeed you've got two images formed in holography the thing is that the one of them is not good and i'll show you later why right so let's go now to the mathematics and i'm saying that we've got one laser beam shown on that er photographic film a laser beam and that's its formula how you define that beam mathematically we call it the reconstructive reconstructing wave or reconstructive beam that's the name for it and you've got this R-subscript there and it's very simple to describe it it's a wave it's got its amplitude and it's got its phase that's what it is now that beam reads the information on the photographic plate that reading is mathematically expressed as er multiplication of the wave the reconstructing wave with the intensity on that photographic plate so what i'm saying is that the final wave the emerging wave behind the photographic plate is going to be a multiplication of the reconstructing wave with the intensity of that that is stored on that photographic plate that's how it is that's the starting point now what's going to happen let's see next bit is okay let's see what happens if we substitute the reconstructing wave with its analytical form and i substituted here the intensity with the formula that i've shown you on the on the previous page nothing new here and what we do is we multiply all that with a constant provided by these two terms and if we do that multiplication we're going to see that it is the reconstructing wave multiplied by this constant and then the second bit of the multiplication is to take all that and multiply it with all that so what's interesting in this multiplication you've got the multiplication of two cosines now the cosine appearing here the cosine there and the result is shown on that second line i'm showing so what do you have here you've got all your constants that you're not particularly worried about at the moment and you've got the cosine of the reconstructing wave multiplied with the cosine of that phase difference between the reference beam and the object right and when we see cosines multiplied together what do we do we use the known trigonometric formula and finally we've got here the f-, the result that is telling you that the final E-F wave that emerges behind the photographic film is the first term that we saw here the reference wave multiplied by constant but that's not giving you any information so basically you're not worried too much as a physicist about that and then you've got the interesting bits you've got the constant here which is the amplitude of the wave arising from the object multiplied by something you're not worried about and you've got the cosine of the summation of these two phase terms do you see what i mean i've taken the multiplication of cosines and i said that cosine-A cosine-B is going to be cosine A-plus-B A being that B being that big term and then i've g-, ended up then with that expression and on the third term which is called the difference i've got cosine of A- minus-B so cosine of all that phase minus all that phase and that's the final wave that emerges behind the photographic plate what does that mean to make it more easy i've written out here again the three terms and we're going to make comments on these three terms so the first term as we said before contains not much interesting information just a constant now the second term which we called the sum term what does it contain we saw here it's got an amplitude which is a modified version of E-nought-nought but what does modified mean well it's been multiplied that's what it is and it also contains the phase here contains the factor two- phi i'll remind you that phi is the phase of the wave coming from a laser now it's getting a little bit complex and trust me on that stage that that term two-phi is responsible for the separation of the two images like i've shown you before the appearance of that term makes the two images that we've seen on the previous figure appear and it also contains information about the phase of the object phi-nought only what's the difference it's got a minus there so that particular term contains information about both the amplitude of the object plus the phase and somebody may say oh okay i've got my amplitude i've got my phase i've got my three-dimensional image no it's not quite right because the phase appears here with a minus what does that mean in practice that that particular image that we called the real image is not right it's upside down so if in reality let's say er something is supposed to be closer to you on that image that bit looks to be further away than you it's the other way around that effect is caused by this minus- phi and the image occurring due to that term is not right is not correct the correct three-dimensional image acu-, occurs from the other term on the final wave which is the difference term and if you see in that difference term you've got the amplitude of the wave coming from the object and now you've got the correct phase phi-nought of the object and that part of the final image is the one that is given you the three- dimensional image in holography so to conclude what we've done today is first thing to remember is the difference between two-dimensional and three-dimensional imaging in two dimensions what we do is we store information about irradiance intensity however you want to put it but we have no information about phase and holography comes and says okay there is a technique where i can store information on both intensity and phase then we shown the basic mechanism of recording and reconstruction in holography and we tried to explain that idea using two ways one way was based on Fourier analysis and we had to say some things about spatial frequencies and the way they propagate in space and the other way to do it was direct mathematical way by using simple techniques of adding waves and calculating intensities and giving physical meanings to various mathematical results and that brings this lecture to an end thank you for your attention and er i'm sure you will all be pleased to hear that there is no workshop today and there is no second hour today so you're free to go home and enjoy your weekend and as about the marks because some people asked me at the beginning of the lecture i haven't done your marking yet i will have by Monday but the gentleman who deals with giving you the marks is Dr namex so don't know when you will know the results of the of the examination you have to ask him that's it